Massive 3loop Feynman diagrams reducible to SC ^{*} primitives of algebras of the sixth root of unity
Abstract
In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3loop tetrahedral vacuum diagram is reduced to 4letter words in the 7letter alphabet of the 1forms Ω:=dz/z and ω_p:=dz/ (λ^{p}z), where λ is the sixth root of unity. Three diagrams yield only ζ(Ω^3ω_0)=frac1{90}π^4. In two cases π^4 combines with the EulerZagier sum ζ(Ω^2ω_3ω_0)=sum_{m> n>0}(1)^{m+n}/m^3n; in three cases it combines with the square of Clausen's Cl_2(π/3)=Im ζ(Ωω_1)=sum_{n>0}sin(π n/3)/n^2. The case with 6 masses involves no further constant; with 5 masses a DeligneEulerZagier sum appears: {frak R} ζ(Ω^2ω_3ω_1)= sum_{m>n>0}(1)^m\cos(2π n/3)/m^3n. The previously unidentified term in the 3loop rhoparameter of the standard model is merely D_3=6ζ(3)6Cl_2^2(π/3)1/24π^4. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for ζ(3) and ζ(5), familiar in QCD. Those are SC^*(2) constants, whose base of superfast computation is 2. Mass involves the novel base3 set SC^*(3). All 10 diagrams reduce to SC^*(3)\cupSC^* (2) constants and their products. Only the 6mass case entails both bases.
 Publication:

European Physical Journal C
 Pub Date:
 April 1999
 DOI:
 10.1007/s100529900935
 arXiv:
 arXiv:hepth/9803091
 Bibcode:
 1999EPJC....8..311B
 Keywords:

 High Energy Physics  Theory;
 High Energy Physics  Phenomenology;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 41 pages, LaTeX